Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups

Miller, Kenneth G.

doi:10.1090/S0002-9947-1983-0716847-2

Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups
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Trans. Amer. Math. Soc. 280 (1983), 721-736 Request permission

Abstract:

Let $P$ be a right-invariant pseudodifferential operator with principal part ${P_0}$ on a simply connected two-step nilpotent Lie group $G$ of type $H$. It will be shown that if $\pi (P_0)$ is injective in ${\mathcal {S}_\pi }$ for every nontrivial irreducible unitary representation $\pi$ of $G$, then $P$ has a pseudodifferential left parametrix. For such groups this generalizes the Rockland-Helffer-Nourrigat criterion for the hypoellipticity of a homogeneous right-invariant partial differential operator on $G$. If, in addition, $\pi (P)$ is injective in ${\mathcal {S}_\pi }$ for every irreducible unitary representation of $G$, it will be shown that $P$ has a pseudodifferential left inverse. The constructions of the inverse and parametrix make use of the Kirillov theory, their symbols being obtained on the orbits individually and then pieced together.

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